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Topic: Help with particle on a ring (benzene)  (Read 3131 times)

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Offline spidermclovin

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Help with particle on a ring (benzene)
« on: January 27, 2017, 01:15:05 PM »
I've been given the question: "show that the wavefunction ψ = (1/sqrt2pi)eimlΦ is a solution of the Schodinger equation (-ħ2/2mr2)(d2ψ/dΦ2) = Eψ and hence derive the equation for allowed energies. I'm fine with deriving the energy equation but I don't know how to prove it is a solution  ???

Offline Borek

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Re: Help with particle on a ring (benzene)
« Reply #1 on: January 27, 2017, 05:10:14 PM »
Have you tried to simply plug the wave function into the equation?
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Offline Enthalpy

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Re: Help with particle on a ring (benzene)
« Reply #2 on: January 28, 2017, 02:40:29 PM »
Check that it's a function. That is, it must have the same value after an integer number of turns.

And that it's a solution: d2(e)/dΦ2 shouldn't be too hard. With all constants.

And that it's normalized: |ψ|2 integrates to 1 over the domain, here 0≤Φ<2π.

The more interesting question would have been: why can we write Schrödinger's equation that way? In Δψ, this writing neglects d2ψ over the radial and polar directions despite the variations are steep, and keeps only the dependence on the azimuth rΦ. The reasons aren't bad, but such lacks of explanations make QM abstract.

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